Let me see what happens if I put in a specific example. Suppose that
If 3*5=35 is a theorem of PA, then 3*5=35
is a theorem of PA. Let me refer to “3*5=35 is a theorem” as sentence 1; “3*5=35″ as sentence 2, and the implication 1->2 as sentence 3. Now, if 3 is a theorem, then you can use PA to prove 2 even without actually showing 1; and then PA has proven a falsehood, and is inconsistent. Is that a correct statement of the problem?
If so… I seem to have lost track of the original difficulty, sorry. Why is it a worry that PA will assert that it can prove something false, but not a worry that it will assert something false? If you’re going to worry that sentence 3 is a theorem, why not go straight to worrying that sentence 2 is a theorem?
We generally take it for granted that sentence 2 (because it is false) is not a theorem (and therefore sentence 1 is false), and the argument is meant to show that sentence 3 is therefore also not a theorem (even though it is true, since sentence 1 is false). This is a problem because we would like to use reasoning along the lines of sentence 3.
To see the real issue, you should replace sentence 2 with a conjecture whose truth you don’t yet know but would like to know (and modify sentences 1 and 3 correspondingly). Now wouldn’t you like sentence 3 to be a true? If you knew that sentence 1 was true, wouldn’t you like to conclude that sentence 2 is true? Yet if you’re PA, then you can’t do that.
Let me see what happens if I put in a specific example. Suppose that
is a theorem of PA. Let me refer to “3*5=35 is a theorem” as sentence 1; “3*5=35″ as sentence 2, and the implication 1->2 as sentence 3. Now, if 3 is a theorem, then you can use PA to prove 2 even without actually showing 1; and then PA has proven a falsehood, and is inconsistent. Is that a correct statement of the problem?
If so… I seem to have lost track of the original difficulty, sorry. Why is it a worry that PA will assert that it can prove something false, but not a worry that it will assert something false? If you’re going to worry that sentence 3 is a theorem, why not go straight to worrying that sentence 2 is a theorem?
We generally take it for granted that sentence 2 (because it is false) is not a theorem (and therefore sentence 1 is false), and the argument is meant to show that sentence 3 is therefore also not a theorem (even though it is true, since sentence 1 is false). This is a problem because we would like to use reasoning along the lines of sentence 3.
To see the real issue, you should replace sentence 2 with a conjecture whose truth you don’t yet know but would like to know (and modify sentences 1 and 3 correspondingly). Now wouldn’t you like sentence 3 to be a true? If you knew that sentence 1 was true, wouldn’t you like to conclude that sentence 2 is true? Yet if you’re PA, then you can’t do that.